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Theorem cmpfii 23237
Description: In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
cmpfii ((𝐽 ∈ Comp ∧ 𝑋 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑋)) → 𝑋 ≠ ∅)

Proof of Theorem cmpfii
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fvex 6895 . . . . 5 (Clsd‘𝐽) ∈ V
21elpw2 5336 . . . 4 (𝑋 ∈ 𝒫 (Clsd‘𝐽) ↔ 𝑋 ⊆ (Clsd‘𝐽))
32biimpri 227 . . 3 (𝑋 ⊆ (Clsd‘𝐽) → 𝑋 ∈ 𝒫 (Clsd‘𝐽))
4 cmptop 23223 . . . . 5 (𝐽 ∈ Comp → 𝐽 ∈ Top)
5 cmpfi 23236 . . . . 5 (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))
64, 5syl 17 . . . 4 (𝐽 ∈ Comp → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))
76ibi 267 . . 3 (𝐽 ∈ Comp → ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅))
8 fveq2 6882 . . . . . . 7 (𝑥 = 𝑋 → (fi‘𝑥) = (fi‘𝑋))
98eleq2d 2811 . . . . . 6 (𝑥 = 𝑋 → (∅ ∈ (fi‘𝑥) ↔ ∅ ∈ (fi‘𝑋)))
109notbid 318 . . . . 5 (𝑥 = 𝑋 → (¬ ∅ ∈ (fi‘𝑥) ↔ ¬ ∅ ∈ (fi‘𝑋)))
11 inteq 4944 . . . . . 6 (𝑥 = 𝑋 𝑥 = 𝑋)
1211neeq1d 2992 . . . . 5 (𝑥 = 𝑋 → ( 𝑥 ≠ ∅ ↔ 𝑋 ≠ ∅))
1310, 12imbi12d 344 . . . 4 (𝑥 = 𝑋 → ((¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅) ↔ (¬ ∅ ∈ (fi‘𝑋) → 𝑋 ≠ ∅)))
1413rspcva 3602 . . 3 ((𝑋 ∈ 𝒫 (Clsd‘𝐽) ∧ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)) → (¬ ∅ ∈ (fi‘𝑋) → 𝑋 ≠ ∅))
153, 7, 14syl2anr 596 . 2 ((𝐽 ∈ Comp ∧ 𝑋 ⊆ (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑋) → 𝑋 ≠ ∅))
16153impia 1114 1 ((𝐽 ∈ Comp ∧ 𝑋 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑋)) → 𝑋 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  wne 2932  wral 3053  wss 3941  c0 4315  𝒫 cpw 4595   cint 4941  cfv 6534  ficfi 9402  Topctop 22719  Clsdccld 22844  Compccmp 23214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-om 7850  df-1o 8462  df-en 8937  df-fin 8940  df-fi 9403  df-top 22720  df-cld 22847  df-cmp 23215
This theorem is referenced by:  fclscmpi  23857  cmpfiiin  41949
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